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Crystal Structure of LixNi2-x02 and a Lattice-Gas Model for the Order-Disorder Transition
B. Sc., Zhongshan University, 1985 M. Sc., Zhongshan University, 1988
A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE
in the Department 0 f
Physics
0 Wu Li 1992 SIMON FRASER UNIVERSITY
March 1992
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APPROVAL
Degree: -Vaster of Science
Title of thesis Crystal Structure of L i & J i ~ - ~ 0 2 and a Lattice-Gas Model for the Order-Disorder Transition
Examining Committee:
Chairman: Proi. E. D. Crozier
12ssijcinte Prof. 3. R. Dahn Senior Super visor
Assistant Prof. ,I. Bechhoefer
Prof. R. F. Frindt
Proi. %I. Plischke Emminer
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ziy thesis, project or extendad essay i the title of which i s shown belo*!
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by me or the Dean of Graduate Studies. It is u n d e r ~ 7 ~ d that c o p y i n g
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T i t l e of Thesis/Froject/Extended Essay
Author:
(signature)
Abstract
The crystal structure uf LixNi2-xQ2 for 0 3 < 1 is determined. For x<0.62,
tAixNi2-xC)2 has the disordered rocksalt structure. For 0.621s_<1.0, Li ar,d Ni
atoms segregate into Li rich and Ni rich layers normal to one of the 4 (111)
directions in the cubic lattice. This breaks the cubic symmetry and an
apparently continuous transition to a hexagonal phase occurs. At x= I ,
alternate cation layers are nearly pure Li and pure Ni. The degree of order is
quantified by defining an order parameter, , which we take to be the
difference in the Li composition of neighboring metal atom layers. Using X-
ray cliffrac~ion and Rietveld profile refinement, we have measured versus x.
We model IdixNi2-x02 as a lattix of oxygen atoms into which cations (either Li
or Ni ) can be inserted, subject to the constraint that the number of cations
ecpals the number of oxygen atoms. Interactions between cations on nearest
and on next-nearest-neighbor sites are included, and the lattice gas model is
solved using ~nectn-field and Monte Carlo methods. We find that the nearest-
neighbor interaction is not sensitive to the observed ordering and that we can
describe the experimental variation of q versus x for an appropriate choice of
the r~ext-l~earest-neigl~bor interaction.
I wish to thank Dr. Jeff Dahn for suggesting the research topic, and tor his
encouragement and advice throughout the entire research work. I would like
to thank Dr. f . R. Reimers for kis very useful assistance during this work. 1
am grateful to my fellow graduate students Hang Shi and 13rian M, WL1y. I
appreciate the assistance and work done by the physics secr&wid m c l
technical staff and by those in the Machinc Shop. I also thank Dr. John
Bechhoefer and Dr. Robert Frindt for serving on the co~nmittse.
Conf ents
1 ntroduction
............................................... 1 .I Rechargeable Batteries
........................................................... 1.2 Li-Ni-0 Materials
3.3 Studies of Order-Disorder .............................................
Sample Preparation
X-ray Diffraction ...........................................................
Structure and Order Parameter ................................................
Refinement Results
5.1 Iiietveld refinement method ....................................
5.2 X-ray Refinement Results ....................................
Lattice Gas Model ........................................................................
Monte Carlo Results ............................................................
Short Range Order ...........................................................
Discussison ........................................................................
Ap2endix .........................................................................
References .........................................................................
V
List of Figures
....... A charge q with acceleration n producing an electric field I 5
An rrnpolarized primary beam along the X-axis sca tiered
by a single free electron at the origin ............................................. 15
A group of electrons scattering X-rays ............................................ 17
X-ray diffraction patterns in powders recorded by the counter ...
diffractomator ................................................................................. 20
Metal positions in LiXNi2-. O2 .................................................... 25
.......... X-ray diffraction patterns for three LixNi2-w02 samples 34
Experiment and fitted profile for LixNi2-x02 with x=0.965 .......... 35
.......... Experiment and fitted profile for LixNi2-x02 with x=0.667 36
Experiment and fitted profile for LiXNi2.. 0 2 with x=0.55 ......... 37
Experiment and fitted profile for LixNi2-s02 with ~ ~ 0 . 4 5 .......... 38
The composition of LixNi2.. 0 2 measured from mole ratios of
reactants ("x from synthesis ") plotted versus the composition
determined from Rietveld profile analysis if*x from Riutveld
profile fitting) ................................................................................. 39
The cell constant. ch. of the hexagonal structure vs . "x from
7 The cell constant, ah of the hexagonal structure vs. "x from
I?i&veld profife fitting" in tixNi2-xfJ2 ............................................ 41
8 The ratio eh/ch multiplied by ?'24vs. "x from Rietveld profile
fitting" .................................................................................. 42
9 The long range order parameter from Rietveld profile fitting
vs. "s fiam Rietveld profile fitting " ....................................... 43
10 The cation FCC frame showing the nearest-neighbor and
next-nearest-neighbors to a particular site ............................... 47
I l a The order parameter versus s calculated by the Bragg-Williams
............................................ model for several values of b /k sT 54
11b The order-disorder phase diagram calculated using the Bragg-
Williams model ............................................................................ 54
12 The FCC cubic structure composed of four simple cubic
la t tires .......................................................................................... 58
13 The order parameter q from the Monte Carlo calculations for
various lattice sizes and a fixed next-nearest-neighbor interaction
cons tan t ........................................................................................ 63
!4 Order parameter, 11, versus s in LixNi2-x02 f r ~ m exoeriment,
mean field and Monte Carlo method .......................................... 64
15n The ilnctuations in the order parameter calculated from Monte
..................... Carlo merages and the Mu1 ti His togram t net hod 65
v i i
. ,
Power law extrapolation of xi to intkite lattice size sI\-ln:;
x, =0.62 ......................................................................................... [I,?
The phase diagram cakufated by Monte Cclrlo method with I.=I2
and J I =-0.5ksT ............................................................................. (76
The order parameter calculated by Monte Carlo method for
........................................ various f with L=12 and f l=-O.Sk~T
The order parameters calculated by Monte Carlo method for
....................................... various 11 with L=12 and ]2=1.35k~T
X-ray diffraction showing the short-range order in Idi,Ni2..xO~
......................................................................................... for 0L~50.62
List of Tables
1 ResuI ts of profif e refinement for LisNi2-x02 samples ........ ........... 44
1. Introduction
1 .I Rechargeable Batteries
Electrochemical energy storage will become more impi-'tar? t as t h t. dim t,!j:c
of fossil fuels and pollution grows more serious- IZc.nc\v,iblt. enrrgit):; s ~ c h , ~ s
solar or wind will be widely used energy sources if ;I more &iicitwt i ~ : t ~ t i i i ~ c i is
found to store their energy. Electrical energy is the nwst conv~nii3nt onc tu
use, and rechargeable batteries can directly change chenrical c ~ ~ : $ r g y in{:>
electrical energy. However, energy densities of commercial succ.tr-tdrrry bC~tft~sii~s
(typically 100 Whkg-1 ) are far below energy densities of nm-renc.\v;ibltr
and polluting fuels such as gas or oil (1?_,000*iYhkg-~). l i the large
difference between [l] theoretical (up to >I000 WItkg-1) and practicd cncrgy
densities of batteries were reduced considerably, then secandary batteries
could be widely used in electric cars and other applications.
The periodic table is the best place to start looking for a suitable material for
batteries. Electronegativity and atomic weight are initial factors to be
considered for high electrochemical energy densities. 'The e1r-lmc.n ts in Ihc
upper left corner should be candidates for the anode and the those in thu
upper right corner (except for the noble gases) for the cathodes witl~ollt
considering cost or safety factors. Although hydrogen is thc lightest aknrn, i t is
not the best element for the anode since solid hydrogen - does riot exist a t
room temperature. -Metal hydrides (e.g. HxM) are used fur the anode.; in Ni-
Metal hydrides cells which are rapidly replacing Ni-Cad cells in thc
marketplace. Lithium is the next material to be considered and it providcs
Sc-co~ilci~trx~ Lithium battcrics have some advantages over conventional
tclchnc~lrvg~cs [2]. They c~11-i operate a t room temperature and offer high energy
density [up lo f OD Wlir:kg fur AA-size cells). They can store energy for a long
time ( u p to 5 to years) ivithout much loss and offer high cell voltage (up to
3.5 volis pcr cull). Eosvcvcr, saietv problems have to be solved before they are
;zridu!y available. Furlhermore their cost is higher than conventional
t e c l l i f ~ l t f g i ~ ~ .
Iiechargeable Li batteries mainly consist of three materials, the negative
dcctrode, tile positi.i7e electrode and the electrolyte. Most eIectrolyte solvents
are chosen from propylene or ethylene carbonate, 2--MeTHF, and DME [6]. The
electrolyte salts are usually LiAsFb, LiCF$O;, LiPF6, or LiiV.'(CF3S02)2. High
rate li thi~tm batteries have very thin electrodes (100 microns thick) because
these liquid electrolytes usually have conductivities about two orders of
magnitude lower than the aqueous electrolytes used in Nickel-Cadmium
cells. Thin electrodes are used to lower the dissipation inside the batteries.
Polymer electrolytes like LiX(CF3S02)2 /polyethylene oxide (PEO) can not
achieve good rate capabilities rvithout extremely thin electrodes since
condwtivities of the electrolytes are txvo orders of magnitude lower than
thost. of liquid nttn-aqueous electrolytes.
Before the scrir~rrs safetr. problem associated with secondary cells was
rt.cognizd, rnetn!lic Li ,trmdes or Li-XI alloys were used as anodes throughout
m a t oi [he I'S;Si_f's. Other alloy systems such Li-Woods metal appeared in
some commercial re%+ After the yery poor safety record was reported [7], new
technologies xvere nut t~scc? in b;tttcri. ti^ iriti-ti>~tr c h , ~ r c h i i ; i
consideration of safetv. Recently, nii-ract1r.e i t . 1 1 ~ li;ii*ts bct.11 tc>unci x s t ~ t > r . c > .t
- 1 second intercalation coinpound is chosen fur the nt.g,lii\~t~ clt.c!r-tjc?i\ {i, -1, :.I
The negative and positive electrodes are both ii~tcrcal,~iioi-i I I U ~ C S ~ I ~ I ~ S rilt~i ill^
Li atoms shuttle back arid forth bet\\-een the electrodes ivltc'n r t w l i <>k1t.Trtftk~
Such cells have been called "rocking-chair" culls. Noxs, low slu-t',~c-e ;rrc,I (1 10
m2/g) carbon is used for the negative elcclrode in these roiLir?.g chair i-t.11:;
There are many materials used for the cathodes o f rc.cfiargcltblc I,i Lx~tt~>ri(>s.
Many new technologies developed rapidly in 1980's because of m a n y ct\oic.t.s
for the cathode and anode and for the electrolyte. On the ather h~rtci, ~ n u c h
work must be done before a new technology can displace old ones. fv1,iny
factors such as safety, good performance and cycle life are important for
practical uses. It is very difficult to find a cathode material which satisfies all
the needs for practical uses; here we just introduce some of the materials used
for the cathodes. Many metal oxides have been studied carefully for- their
electrochemical properties; these include vanadium oxides, c h r o r n i ~ ~ m
oxides, manganese oxides, cobalt oxides and nickel oxides.
Vanadium oxides have been studied extensively as cathode materials. It has
been found that Ei atoms can intercalate into these materials. Some structural
aspects regarding these compounds have been reviewed by Abraham 1181.
Many reports appeared characterizing V205, LiV308/ V6013 [9/ 10, 12, 241. X-
ray diffraction, electrochemical and infrared techniques arc. common
methods to investigate these materials.
Several steps in the voltage-composition profile occur during discharge of Li
cells with crystalline V205 cathodes. Above 2.0 V, three charge ~ 'q~~ivc l l e~ t t s
c:trri:ir;::r~c!iizg tr, l,i3V2il15 can bc provicicci, which corresponds to a charge
c.icrisi!.y (?I 440 i l h k g - i . i"heorc.iica1 energy densities u p to 650 Whkg-1 are
achieved during iictw discharge for primary t i / V 2 0 s cells [9]. Hoivever, good
cycling beiiavirir is observed only if s in L,isV205 is limited to 1-1.5 [lo]. I t has
ttccn confirmed 1291 that '$205 is very sei~sitive to overdischarge and that V - 0
bo Js arc broken fr;r x i l .
1-il irt7-108 W ~ S introduc-ed as a cail~acie material for Li cells by Nassau et al.
irt I981 [I:]. Every Z,iV308 unit ran accept up to three more lithium atoms by
cheinical iiitercaiaiion with butyllithium [12]. Li/LiV30q cells have a
iheorctical charge density of 280 Ahkg-1 and a specific energy of 680 Whkg-1.
Its Ltpplication is limited by the disadvantages of low electronic conductivity
and high oxidizing capability which may result in electrolvte decomposition
tm
V6013 was used as a cathode material by Murphy et al. [14]. Much
st~bsequent work concentrated on the electrochemical characteristics of V6013
and its performance in secondary Li cells with organic electrolytes. Each V6013
unit can acco~nmodate up to 8 Li with butyllithium when
nonstoichiomctric. The highest capacity is obtained for 2.17a<2.19 in VO,
[ltil. V6013 (n=8) has a large theoretical charge density (420 Ahkg-I), but only
six Li can be inserted reversibly in V6OI3 [16]. The discharge is divided into
three rnain steps corresponding to the consecutive transfer of one, three, and
four electrons. V 6 0 1 3 is practically insoluble in organic solvents in
comparison to V20_5. This gives a low self-discharge rate and good shelf life.
Chromium oxides, CrO, have been studied as cathode materials in the range
22~12.67 [I?]. Yarnarnoto et al. reported 1203 that amorphous Cr308 is better
Mn02 is the most popular cathode ~mierials for gcdvmic cells sinw i t is
cheap and has reasonable capacity and electrode potential. Currently, cbigl-11
billion primary zinc/manganese dioxide batteries are produced each yens 1211.
Mn02 has been investigated as a cathode material in p r i m ~ r y Li cells since thc
1970's [22]. Sony used to produce secondary Li batteries based on MnU2 1231.
These AA cells offered very high practical energy densities of 125 Whkg-I and
240 Wh/l at discharge rates of C/2.5h (discharge the entire capacity in 2.5
hours) with an average voltage of 2 . W . It is claimed that these cells c m
achieve 1000 recharge cycles. Self-discharge is low at about 1% per month [23].
However, there are still some serious safety problems to solve before these
cells can be really successful products. Moli Energy's Li/Mn02 cells were
similar in performance to the Sony cells.
Nickel oxides are another successful cathode material in commercial
secondary cells. Ni/Cd cells are now used world-wide and their salcs h a w
reached US $ 1.3 billion per year. 'Gickel oxide has been considered as a
cathode material in alkaline cells since 1837 [24]. Even though some ~ ; ; t t ~ ~ n l s
appeared in 1900/1901, their electrochemistry is not yet fully understood .
The open circuit voltage of Li/Li,Co02 cells is as high as 4.7V for x-0.07 [251.
Theoretical energy densities of 1070 Whkg-1 can be obtained based on x=l and
ti^ ;ivc.r;i;;c discharge ~ ; ~ i t ; l g e o i 3.9V. Bil t the high discharge voltage and
r,xjtii~ing poivtr of the Li,CoOz can cause the decomposition the electrolyte,
iorning polymeric films on the electrode surface [26] and corroding the
~il;cicriying metal support. Xeverthelecs, at least one company (Sony) is now
cxpluiting the highly promising clzaracteristics of LiCo02 for practical
reclznrgeable ha tteries.
-Many companies are working hard on rechargeable Li batteries. Early in the
1980's, Moli Energy Ltd. in Burnaby, B. C., developed the first commercial
AA-size rechargeable Li battery with 300 cycles. The cell had thin electrodes,
using Li-125pn and MoS2 on A1 foil as electrodes. These Li/MoS2 AA-size
cells stored about 1.4 Wh of energy and weighed 21 grams. However, several
safety incidents involving the cells [27] forced their withdrawal from the
market. T,i/Mn02 AA-size cells were announced in the late 1980's by several
companies (Sony Energytec, Moli Energy Ltd. and Matsushita). Those
constructed at Moli Energy stored about 2 Wh of energy, weighed about 19
grams and had a 200 cycle life time. Because of safety problems, these cells
were never introduced in the marketplace. On the other hand, the safety
problems in ii-alloy/Mn02 coin cells are rare since the cells are very small.
Now it is understood that the poor safety of rechargeable Li electrodes is
caused by the large area of contact between the cycled Li and the electrolyte.
This unstable surface becomes larger and larger when the cells are cycled and
thermal and/or electrical abuse results in the unacceptable situations.
Apparently, if the surface area of the negative could be kept small or if less
active materials were used, the poor safety could be improved.
In the rocking chair approach, both liegatis-t. , ~ i z c i 17iwi:i~ct c i t~c t ro~ i r~s ;?ri>
intercalation cornponnds ivith high reversibilitie.;. The .;ur-i,\w ,ISPA I.)( thi.
powders used as electrodes is kept sn~zl l daring c),clirzg to p l ~ s ~ ~ r . t t :Ilt\
conditions for safety. Sony has fabricated LiCoC12/Cc>rbon rc>c i~ ,~rgc ; \b l~
batteries in a variety of sizes. The AA size cells can store ;~buu t 1.4 W h c:nc,rgy
and be cycled up to 1200 times. Many other conzpanies arc no\-v conc.entsc~ting
on the rocking chair approach to construct rechargeable Li battcrics.
LiNi02 is another choice for the cathode. Many studies 1 ~ v c been mCtdt~ to
understand rile electrochemical characteristics of Li-Ni-0 nlaicrinls. Since this
thesis concentrates on LixNi2-x02, it is very rrselul elaborate nzorc a1w~1t ~ v o r k
done on that material.
1.2 Li-Ni-0 Materials
NiO has the rock-salt structure. Li-Ni-0 compounds usually have one oxygen
atom per metal atom and are normally based on layers of close-packed oxygen
atoms . LiNi02 was first prepared by Dyer et a1 [30]. Goodenough et al. [31,32]
and Bronger et al. [33] studied the phases in the solid-solution series
LixNi2-x02 (O<x<l). They showed that a disordered rock-salt structure exists
for ~ ~ 0 . 5 6 and that partial cation ordering occurs for ~ 0 . 6 . They 5uggcslc.d
that there is an ordering of the cations in the alternate ililjc ( ( i l l ) in the
cubic structure) planes and calculated the order parameter as a function of x
based on the magnetic measurements and the analysis of X-ray diffraction
patterns.
- i.iNiOn hds the space group R 3 m with al,=2.876A and cj,=11.19A. Other
Ti-Ni-O compounds exist besides LiNi02 . Additional lithium can also be
inserted electrochemically into LiNiOz to form a new Li2Ni02 phase (IT-
Li2NiOz ) with the Ni(OH)2 structure [28]. When the additional Li is removed
from T3i2NiO2, the host lattice reverts to the LiNi02 structure, but there is
substantial hysteresis in the voltage of Li/LiliyNi02 cells during this phase
transition. The analogous phase, Li2Co02, does not form when additional Li
is added to LiCo02 by electrochemical methods. The transformation
between these phases can be made by displacing 0-Ni-0 sheets normal to the
c-axis. In the 3R-LiNi02 structure, all cations are octahedrally coordinated by
oxygen and the Ni layers adopt ABCABC-stacking to maximize the distance
between cations. When L i 2 N i 0 2 forms, additional lithium must be
accommodated within the lithium containing layers. This necessitates a
filling of the tetrahedral sites accompanied by a displacement of the 0 -N i -0
sheet so that the Ni layers exhibit AAA-stacking; this maximizes the distance
between cations . This phase has the space group P 3 ml.
The orthorhombic phase of Li2Ni02 was reported by Rieck and Hoppe [34].
who found a=3.843A, c=2.779A. The IT-Li2Ni02 phase is changed to the
orthorhombic phase when heated to 500•‹C.
Room-temperature electrochemical extraction of lithium from nearly
stoichiametric, layered LiNiO-, gives the solid solution system Li1-~Ni02 .
The normal spinel Li[Ni2;O4 is formed by a 200•‹C anneal of the metastable
layered roil~pound Lip5Ni02 obtained by electrochemical extraction lithium
from LiNi02 at room temperature [35].
In addition to structuraI st~ldies, lithium nickel oxides have been studiccl. as
catalysts for the direct conversion of ~nethane to higher hydrocarbons
through oxidative coupling E361 and for other oxidation reactions such as C-O
oxidation [37]. The selectivities observed for oxidative coupling oi methane
are comparable to those of other catalysts such as LiJMgO 1381. The gorsd
electronic conductivity also allows the use of iithium nickel oxides as an
electrocatalyst for methane oxidation and improved selectivities have been
reported under conditions where oxygen ions were eiectrocheinically
pumped away from the oxide catalytic surface. Electroca taly tic and oxygen
exchange kinetic data indicate that oxide ion diffusion is significant, but little
quantitative information is currently available
The "rocking-chair" rechargeable batteries that use lithium intercalation
compounds for the positive and negative electrodes are safer than batteries
that contain free-lithium metal because of the unstable character of lithium
metal. LiNi02 [4, 281 has been proposed as an electrode material in secondary
Li batteries, because Li can be electrochemically de-intercalated from this
material to form LildyNi02. LixC6 is used as the anode which has a cher-nical
potential close to that of Li metal. Cells with LiNi02 as the cathode with
petroleum coke as the anode have been made, and they have high energy
density, long cycle life, excellent high-temperature performance, and low sclf-
discharge rates. Furthermore they can be repeatedly discharged to zr2ro volts
without damage, and are easily fabricated.
The performance of the cells depends on the stoichionwtry ct i I,i,Ni2 ,02
used in the cell. hTi atoms enter the Li layers and presumably these hJj atoms
impede the diffusion of Li. Therefore, it is important to know how much Ni
enters the 1,i layers as a function of x, so i t is necessary to have a careful
structural study on this solid solution.
1.3 Studies of Order-Disorder
X-ray diffraction is one well developed method to study order-disorder
transitions. Binary alloys, such as J3-CuZn, are typical examples where order-
disorder transitions have been investigated by the X-ray diffraction methods.
Review articles include those by Nix and Shockley [39], Lipson [40], Guttman
[41] and Gerold [42]. LixNi2-x02 is not a binary alloy ( it is a pseudobinary), but
its order-disorder structure can be studied in a way similar to that which has
been s~lccessful on the binary alloy of P--Brass.
The P brass phase of the copper zinc system occurs in a small range of
composition near the equiatomic composition CuZn. The atoms are found in
the ordered structure if brought to equilibrium at low temperature. In the
ordered structure, the cube corners are occupied by the copper atoms while
the cube center positions are occupied by the zinc atoms as in the CsC1
structure typr. This is a body-centered-cubic structure. At somewhat higher
temperatures, the structure is not perfectly ordered, but most of the copper
xt. still on thc cube corners and the zinc atoms are stiii on the cube center
p~Gtittits. The long-range order still exists, but i t is only partial long-range
urdcr. Thc critical temperature is around T,=460•‹C depending on the exact
rv~nps i t ion . Thc 10"s-range order disappears above this temperature, and
the corner and center positions are occupied randomly by c o p p c r a ~ ~ d zinc
atoms. Although long-range order vanishes above the critical tempera turc.,
short-range order still exists. This means that the probability of occupation of
certain positions are affected by the occupation of close-neighbor positions
but not by the more distant positions.
For long-range order in binary compositions, there are two kinds of atoms A
and B and two kinds of positions, a-sites and p-sites. Perfect long-range
order is defined as that which occurs when the a-sites are all occupied by A-
atoms and the p-sites by B-atoms. If N is the number of atoms in the sample,
NA=XAN and N B = x ~ N, such that x ~ + x g = l , where x~ and xg are the atom
fractions. If ya and yp are the fractions of a-sites and p-sites, then y,+yj =l.
The number of each kind of site is given by N,=yaN and Nj=ypN.
Nix and Shockley introduced four important parameters: r , for fraction a-
sites occupied by the right atom; w, for fraction of a-sites occupied by the
wrong atom; rp for fraction of p-sites occupied by the right atom; ili,j for
fraction of p-sites occupied by the wrong atom. Suppose each site is occupied
by one atom and these are no vacancies. Two relations bclwcen thc:
parameters are immediately obvious:
r,+zu,= I and rp+zop= 1,
since the fraction of the sites occupied by A-atoms equals the fraction of A-
atoms and the fraction of the sites occupied by B-atoms equals thrk fraction of
B-atoms. There are two equations, which are easily obtained
It is convenient to define the Bragg-Williams long-range order parameter q
for nonstoichiometric compositions that is linearly proportional to (r,+rp)
with q=0 for a completely random arrangement and q=l for the completely
ordered stoichiometric phase with r,=rp=l. Then the long-range order
parameter is expressed in the simple form:
The parameter q can not reach its maximum value of unity except for a
stoichiometric composition with perfect order-range order. With such a
definition of order parameter, the structure factors for nonstoichiometric
compositions are proportional to q for the Miller index related to this ordered
structure. The order parameter q is thus easy to observe in X-ray experiments.
2. Sample Preparation
Li2C03 and NiO are usually used as the starting materials to prepare Li,Ni2-
x02, When Li2C03 and NiO in the proper ratio are ground and heated
between 650•‹C and 850•‹C, the Li20 resulting from the decomposition of:
Li2CQ3 combines with NiO to form LixNi2-x02. This ~nethocl is good for
small x . However, since pure Li2C03 only decomposes at high temperature
(800•‹C) , we need more processing to form pure LixNi2-x02 with .r near 1
because of the remaining Li2C03 phase. At higher temperature there is a loss
of Li due to volatilization of Li salts.
Here LixhTi2-x02 samples were prepared by reacting stoichiometric mixtures
of LiOH@H20 and Ni(OH)2 . The powders were ground with a mortar and
pestle and then heated at 700•‹C for 2 hours in air followed by slow cooling
over several hours to room temperature. LiOH reacts rapidly with NiO,
(which forms when Ni(OH)2 deco~nposes around 300•‹C), which mir~i rn izc~
the Li loss due to extended heating that commonly occurs when I,izC1(1)3 i:;
used a starting material. Li2C03 appeared when the 1 , i N i ratio was r w w
than 1. This could be avoided by carrying the reaction c m t undcr 0 2 (I$-)
forms instead), but still we could not prepare single phase matc.iri~ls with
x > l ,
3. X-ray Diffraction
There are three factors which are considered in calculating the intensity of X-
ray diffraction: scattering from electrons , scattering from an atom and
scattering from crystal structures.
The results of scattering from electrons are derived from electromagnetic
theory. An accelerated charge radiates, and the electric field produced by the
accelerated charge is given by
qasin a E =
C' R
where c is the velocity of light, a is the acceleration of the charge q, R is the
distance between the charge and the observation point. u the angle between R
and a. Figure l a shuws the directions of a , R and angle a. The observation
point is P.
Figure l a In cIassica1 elcctrumagnctic theory, a chargc q with
acceleration ti produces the electric field E
PVe cclnsider a sing'se free electron at the origin of figure l b with an
unpaiarized primary beam directed along the X-axis. We choose one
clirection for Eg as shown in figure l b and then the amplitude of the
unpoiarized beam is summed over all directions. A force is exerted on the
eleciron which produces
Using (1.11, we have
e2 E,, Ey. = - COS @ n c 2 R
where the relation between E and E is that
E,, = E:,, sin 2 xvt E,. = E,, sin2nvt
E,, = E,, sin2xvi E,. = E,. sin 2nvt
The resultant amplitude E at the point of observation is given by
Since the incident radiation vector lies in any orientation in the YZ-plane
iviih equal prnbctbility, the Y-axis and Z-axis are equivalent and
TLw intensity I is proportional to <E'>, and the scattered radiation intensity is
$ ~ V L ~ I I by
and (l+cos2$)/2 is called the polarization factor.
Figure l c Scattering by a group of electrons a t point rn.
The wave length of the diffracted X-ray is comparable to thc size of a n atom.
The phase difference from different parts of an atom must be considcrcd.
Figure l c shows the scattering by a group of electrons. At position r,, so and s
are the unit vectors of the direction of the primary beam and the direction of
the point of observation P respectively. In terms of a wave frnfl through 0,
the instantaneous value of the field at point rn is given by:
The total path is X l + X - , , the magnitude and phase of the scattered beam at P
d u e to electron 71, is
Here a 180' jump in phase in the scattering is dropped: it doesn't affect the
final results since it is the same for all the electrons. Using the usual plane-
wave approximations with X2 --->R , we get
In terms of the complex exponential, the sum of the instantaneous fields at P
is given by
where the sum over n is for all the electrons in the atom.
If each electron is spread out into a diffuse cloud of negative charge.
where pi, is a charge density for electron n in space. The scattering factor per
electron is given by
/, = j(,'2~."""." pi1 b'
I i the charge distribulicsn has spherical symmetry for each electron in an
atomf the scattering factor for the atom lvith several electrons is
If we consider the intensity scattered by a small crystal, intense peaks occur
only when the Bragg laiv is satisfied. Since the planes of a @\.en Miller ini1c.u
are spaced on the order of the wavelength of the X-ray, the phase diffcrsnw
from scattered X-rays from the planes must be considered. The structure I;l~ti>~.
is defined as
where h, k, 1 are Miller indices and x,, y,, z, are the fractional coordinates.
This structure factor does not include thermal vibration parameters.
In powder X-ray diffraction, the ideal powder consists of an enormous
number of very tiny crystals of size 10-3cm or smaller with totally random
orientation. These samples can be obtained as fine grained yrecipi ta tes [43], or
by grinding coarse crystalline material, or by making filing,; in the case of
metals. Here we use Li,Ni2_,O2 powder ground with pestle and mortar.
For a powder sample, the large number of small crystals is distributed in all
orientations xvith the same probability. Thus, there is a n u i n h e r o f
orientations for a small crystal in which X-ray scattering can satisfy the
Bragg condition at the same Bragg angle; these orientations correspond to n
certain Miljer index, and the number oi orientations is the mu1 tipiici t y.
X-ray diffraction measurements lvere made using a Phillips prrwdt:r
diffractometer. The radiation used for the powder method is the CrrK c/.lr/.;!
double:, where X1=2 . X E 6 A and k2=1 .53.1.04~.
The diffractometer is illustrated in figure Id . The sample is a f l a t - i a c d d a b
of po-r\~der. Monochroma tic radiation front the source goes through t l i r '
entrance slit E, and strikes the sample. The diffracted radiation is selected by
the detector through a narrow receiving slit R. The distances EO and OR are
made equal, and the sample face is maintained symmetrical with respect to
the primary and diffracted beams. The pulses produced in the detector are
amplified and fed to a computer. During the measurement the detector is
fixed at a particular angle for some time, and there is no measurement while
the angle is changed- The sample face is kept symmetrical with respect to the
primary and diffracted beams because the detector turns at an angle 28 while
the sample t~lrns at angle 8.
X-ray
Sample
Fipirc ti1 X-ray diffraction pattprns i n powders are recorded by the counter diffractometer
For pilu~cier patterns recorded by a po~ rde r diffractometer, the handling - of
the systematic error can be complex. There are systematic errors even with a
wll-buil t diifrnvtometer having a very accurately graduated scale. They
trstmtlv rcsult from three effects:
1. The beam penetrates belotv the sample surface.
2. There is a zero error in the setting of the 28.
3. Axial divergence error occnrs.
Error 1 is of importance for samples ivith low absorption coefficients, but
LixhTi2-x02 is not this case. Error 2 is solved by adjusting the zero point i n the
Rietveld fitting program [a]. Error 3 consists of two parts. One is a constant
term dd/d=h*/18~2) , where d is the distance of diffracting planes, R is the
distance of OR in figure 1.3, andh is the size of the entrance slit. The other is
relative to cos2B and it can be eliminated by a cos*$ extrapolation. Thc
constant term h * / ( 8 ~ 2 ) is about 10-6 and it implies an error h i / d about the
same order.
Thermal vibration affects X-ray diffraction. The most familiar effect of
thermal vibration is the reduction of the intensities of the crystalline
reflections by the Debye temperature factor exp[-?MI. It gives a mcthod for
obtaining the mean square amplitudes of vibration of the atoms <ps2>. Sinrc.
it weakens the reflections that we desire to measure, we ~ n ~ i s l correct for i t in
crystal structure determinations where quantitative values of the structure
factor are necessary. It is easy to understand that vibrations productl a dif'ftlsc
intensity and reduce the intensities of the crystaliine rcflcctions 'rhra
intensities are reduced by a factor exp[-2M] for a crystal containing only oncL
kind of atom. Fcr the general crystal conkaining more than on^; kiiid of a tom,
one replaces each J,; in the structure factor by f,,exp[-Mn]. The fai!c:ie, M,, i ~ ,
given by
where <p,2, is the mean square component of displacement of a t ~ m 72
normal to the reflecting planes- For simple structures, the value of the factor
exp[-2M] can be obtained directly from the measured integrated intensities,
and the corresponding values of the root mean square components of
displacement
4. Structure and Order Parameter
The compounds LixNi2-x02 are structurally related to NiO which has n
rock-salt structure with ac=4.168 A. In this structure, the metal ions occupy
one Face-Centered Cubic (FCC) frame while the oxygen ions occupy the other
FCC frame. These two FCC frames are shifted by a,/2 along sne of the three
cube axes. For 0.02x20.62 LixNi2-x02 is cubic and for 0.62L-..c<1.0 it is
hexagonal. Fig. 2(a) shows the metal atom positions when s = I and cl, and a[,
are cell constants in the hexagonal structure. When c l , / ah=2G the unit cell is
equivalent to cubic one with a,=@ ah.
As figure 2a shows, the metal atoms segregate into predominantly I,i filled
layers (L layers) and predominantly Ni filled layers (N layers) nuar x = l . As x
in LiXNi2-,O2 decreases, Ni atoms must move into the I, layers and T i atoms
may move into N layers. We define a long-range order parameter a s
and where < x ~ > and C C N > are the average .- L,i concentrations in the I , taycrs
and N layers respectively. In figure 2a, q = l since the structure is ycrfectiy
ordered. Figure 2b shows a situation for x < l where thc lattice is partially
ordered and has q<l . Finally, figure 2c shows the cation positions in N i 3
which has q = O according to our definition.
Profile refinements are methods that minimize the difference between
experimental and calculated intensities to obtain structural information. In
our profile refinements, we refine the concentrations X L and x ~ . In an L layer
each site is filled by an average atom consisting of x~ Li and (I - xL) Ni. In an
ft' layer, each site is filled by x~ Li and (1 - XN) Ni. Since lithium has only
three electrons, we are most sensitive to lithium through the absence of Ni,
since we assume the number of cations in the structure equals the number of
oxygen atoms.
Fig. 2 Metal Positions in Li,hTi2-,O2
(a) s=1.0 a r d q=I.O, LiNi02 with the perfect ordered structure.
(b) O < s < l , and O q < l with the partial ordered structure
(c) x=O.O L ~ n d qz0.0, X i0
5. Refinement Results
5-1 Riet-veld refinement method
The Rietveld structure refinement procedure was first introduced in 1966
[45]. The method is widely used in X-ray and neutron diffraction. It is for
crystal structure refinement which does not use integrated powder diffraction
intensities, but employs directly the profile intensities obtained from step-
scanning measurements of the powder diagram. Fourier techniques assist the
Rietveld method when used in the nb irzitio solution of crystal structures
from powder diffraction data.
The first program in Rietveld method was written specifically for the
analysis of neutron diffraction data from fixed-wavelength diffractometers by
Rictveld [2?69]. It ivas developed further by Wewat [I9731 and used successfully
by a number of other workers. The method has been extended to the
refinement of neutron patterns recorded with time-of-flight diffractometers.
The program for the analysis of X-ray data appeared in 1977, and a more
advanced program Ivas used to solve crystal structure problems subject to
qui tc sophistimted constraints.
The program for the liietveld refinement lvas dessrikwl. b-t. t\;ilp,i ,~l;tt 1 i311ng
[46] in analysis both for X-ray and neutron diifr,~rtion p;ttttbrns rcCorticd ~ v i t h
conventional (fixed wavelength) diffractometers. Thc ststlctrrrcs ~ > i t ~ v o
coexisting phases can be refined with this program ~vhich also accom1noc1;ltc.s
data recorded for two wavelengths (such as the X-ray doub1t.t). TIitl.
background can be refined or prescribed in the prosrant as rcquircd. Thclr' is
a built-in table of X-ray scattering factors, and neutron nuclear scdtlt~sing
lengths , and calculation of symmetry operators from a stanciard sp,~w-group
symbol for easy use. The new program [MI tailed LHPMI I V ~ S d w c l ~ p c d by I?,
J. Hill and C. J. Howard and is used here to analyze the X-ray dnia.
The basis for the Rietveld method [44] is the equation
where Yic is
background
the net intensity calculated at
intensity. In our calculation,
point i in the pattern and Yjb is the
Yib was chosen as A+HO+C/N and
A, B, C are profile parameters. G i k is a normalized peak profile function. l k is
the intensity of the kfh Bragg reflection and k is summed over all reflections
contributing to the intensity at point i.
Based on the introduction of Chapter 3,, we give the expression of intensity
I k .
where S is the usual scale factor, M k is the multiplicity, tk is thc Lurc~~t;;!
polarization factor, Wk is an isotropic Debye-Wailer factor which we have
nssirrnod is the wme for cach atom in the unit cell, and FI; is the structure
fnc!or
1v11~rr \, is [he scatiering factor of atom j , and ilk, are matrices representing
the iMiller indices, and atomic coor6inates respectively.
T11c ratio of the intensities for the two wavelengths comes directly from
qtmntum rrtecfr~anics and is equal to 1/2, so that only a single scale factor is
required. The peak profile function Gik is chosen as the pseudo-Voigt
function
c;', cF C,, = y-[f + cox: j-' + ( 1 - y ) ~ q j [ -c] X: ] HL 4 (6.1)
where C p 4 , C/=41n2, Hk is the full-width at half-maximum (FWHM) of the
ktil Bragg reflection, Xik=(2 8i-2 Ok)/Hk, and y is a refinable 'mixing' parameter.
The pseudo-Voigt function can be assigned a fixed shape of any type between
its limiting Gaussian and Lorentzian forms (y=O and 1 respectively for the
pseudo-Voigt). The peak shape can be varied across the pattern by application
of the fimction
where yl, y3 y_? are refinable parameters.
For this profile tvwe, , I the variation of the peak FWHM is defined by the
function described by Gaglioti cf nl. [47].
v~i-rcrti Ip , , is i . h ~ 'observed' iniegraied intensity of reflection k calcuiated at the
i.ttd c i f rciinemr~nt after apportioning each Yjo between the contributing peaks
m d background according to the calculated intensities Ikc.
In the Iiictvcld refimmertt, the values for the X-ray radiation wavelengths
are =I .56056A and hz=1.54440A, and the intensity ratio of IL1 and h2 is exactly
0.5. The width (rangej of calculated profile (in units of Hk) beyond which it is
set !G zero was 3. The coefficient in formula for polarization correction is
0 . H W due to diffracted beam monochromator. The individual isotropic
temperature factor and anisotropic temperature factors were not used. Only
an overall isotropic temperature factor was included in the refinement. The
number of parameters was between 10 to 15 and the number of the data
points was between 1400 to 1900.
5.2 X-ray Refinement Results
Figure 3 shows X-ray diffraction profiles for several LixNi2-x02 samples, to
show the changes in the profile which occur as x changes. The ( 0 0 3 ) ~ peak
decreases in intensity as x decreases, becoming weak and broad below x=0.62,
indicating short-range instead of long-range correlations. The sharp peak
~ ~ p p e a r s only when the average composition of the L and N layers differs. The
raiia of the intensities of the (101)1-1 and the ( 1 0 2 ) ~ ~ ( 0 0 6 ) ~ peaks changes with
.t- 31td itas bee:; pre~iously used to estimate x in LixNi?-xO~ [28].
Figure 4a shows an example of a Rietveld profiie refinement for one of our
i;;anq-&s- For sN.62, the fits included at least 12 Bragg peaks while for ~ ~ 0 . 6 2
only 6 (including 2 pe'iks 'bet~wcn 10 :tnci It)O 20) c~ .~~c~i i rnc~~i i , t i l~ . t n ~ \ ~ i b ~ l r t ~ i
peaks could be used. For ~4.62, the ii~etvriti pr.ogr,lm is L I I L I ~ I C tp f h ~ f i t tht)
broad short-range order peak i w l l w a r 28 =1$.5", bcc'i~~se its ~\~icith docs not
follow the functional form used by the I?iet~.t.ici mt.tltoci. No\-erl'iii.lt.ss, ire.
refined the entire profile and considered the short-rmgc orcicr pc;lk s t 'p; t r ,~t~~ly
as we describe later. Figure 4b sl~o\vs the I i i ~ t v e l d proiile rtliiilcmcni \\,it11
x=0.667 which is above the order-disorder structure transition, !lw pt\;tL near
26=18.5"is fit well. Figure 4c shows the results ivith s=0.55 just ix~lotz~ t l ~ i l
order-disorder transition, the peak near 28-18.5" ~ C C O ~ ~ ~ U S broilti r~lid the
calculated peak can not follo~v its ~vidth. Figure 4ci sltotvs the rcs~tlts tsith
x d . 4 5 which is in the disordered phase f a - from the transition point; tlw
peak near 28=18.5" in experiment is very broad and weak, the cdcalatrd peak
is weak too but the width is narrow. Table I reports the results of our
refinements. For all refinements, Rg, was less than 3.4%, indicating g o d
agreement between the data and the structural model.
The composition of the samples are determined in two ways. First we simply
use the mole ratios of the starting materials mixed in the synthesis. 'The
Rietveld program gave a second v a l ~ ~ e for s, obtained by summing tlw x~ and
XL returned by the program. We call the first composition "x from synthesis"
and the second "x from Rietveld profile fitting". Figure 5 shows that these
compositions agree well with each other, rvhich gives us confidence both in
the quality of our samples and the refinement method. The solid line in
figure - 5 shows the line "s from synthesis"="s from Iiietvrld prclt-lc fir!ing".
Apart from 2 points, our data fail systematically to one side of the firic
suggesting some small problem with our refinement methods or in the
original stoichiometries of our starting materials. In what follows, tvtr plot
rcfincd q u n i ~ i i t i c s ::ursai "s from !Getveld profile fitting" for consistency.
.I I 4tibsti tuking 3: frorn 5ynthesis" does not change the results significantly.
All Iiictvcld profile fitting is based on the hexagonal structure, ever, if the
strtlcturc could be fitkcti by a cubic structure for 0.0<x<0.62 (The (003)H peak
will disppear in calcutation if it is fitted by a cubic structure while the broad
2nd \$7?icak (003)1-i peak exists in experiment). The cell constants al, and cl, in
figure 7 a n d figure 6 are i n good agreement with the data given by other
workers 131, 281. For nr<U.62 in the cubic structure, the refined s differ by less
t21an 0.03 between the cubic and the hexagonal structure used in the Rietveld
profile fitting. Figure 8 shows a al,/cl, from Rietveld profile fitting in the
liexngonal structure, for 0351 . When m a l , / c 1 , = l the structure is cubic. The
error bars are larger for ~ ~ 0 . 6 2 than those for 0.62<x<I.O, because fewer Bragg
peaks are included in the refinements. In our estimate, the composition of
tho order-disorder transition is best obtained from figure 8. We measure
s,=0.62.
Figure 9 shows the order parameter q versus "x from synthesis". At x=l, full
order is still not obtained since q=0.96; this means there are some Li atoms in
Ni rich layers and some Ni atoms in Li rich layers even for stoichiometric
L iNi0 . l . This result agrees with that frorn recent neutron scattering
measurements on L iN iQ [55]. For s<0.62, there is still short-range order as
evidenced by the ureak and broad peak in the ( 0 0 3 ) ~ position. Our Rietveld
refinomcnts are all done in the hexagonal system and do not treat the short
range order peak well. The refinement gives non-zero long range order
parameters helo\\- s,=0.62 in an attempt to fit the weak broad peak. We have
sketched a guide to the eye in figure 9 which indicates how the long range
circler parameter must behave. \Ve have indicated with squares, those fitting
results xvhich are an artifact of oui , ~roiciiurc ;lnd not indicati.\-t. oi in it^ 1t.rrig-
range order. More careful fitting of the iriticni scattering is r ? ~ e d t ~ d t o f ~ t ' i i \ r
understand the short-rmge order.
2 0 3 0 4.0 5.0 Scattering Angle (degrees)
Fig. 3 X-ray diffraction patterns for three LixNi2-x02 samples with x as
indicated in the figure. Miller indices for Bragg peaks referred to in the text
are indicated.
Experiment
J I I\ h A
1 0 0 0 1 Difference
1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 Scattering Angle (Degrees)
Figure 4a Experiment and fitted profile for Lio.96jNil,03j0z as indicated in the
figure. The difference between the two profiles is also shown.
-1000 I I I I I 1 1 1 0 20 30 4 0 5 0 6 0 7 0 8 0
Scattering Angle (Degrees)
1000 0
Figure 4b Experiment and fitted profile for Li0.667Ni1.33302 as indicated in the
figure. This sample is in the ordered phase.
- Difference - - Y
5 0 0 0
h 0 - w .- cn
5 0 0 0 a +--,
s - 0 2 5 0 0
0
1 0 2 0 30 40 5 0 6 0 7 0 8 0 9 0 100 Scattering Angle (Degrees)
Figure 4c Experiment and fitted profile for Li0,55Ni1.4502 as indicated in the
figure. This sample is in the disordered phase.
r -4
~ = 0 . 4 5
r - Calculation -
-
- A A A
I
t 1 Experiment ]
- - I - JL , \ A A.
- Difference -
.i J I
1
I 1 1 I I I I 1 J
Scattering Angle (Degrees)
Figure 4d Experiment and fitted profile for Li0.~5Ni~.5502 as indicated in the
figure. This sample is in the disordered phase.
0.1 0.3 0.5 0.7 0.9 1.1 x from Rietveld Profile Fitting
Figure 5 The composition of Li,Niz-,Oz measured from mole ratios of
reactants ("x from synthesis") plotted versus the composilioil dc termined
from Rietveld profile analysis ("x from Rietveld profile fitling"). The solid
line is "x from synthesis"="x from Rietveld profile analysis"
Fig. 6 The cell constant, ch, of the hexagonal structure vs. "x from Rietveld
profile fitting" in LixNi2-x02.
Fig. 7 The cell constant, a], of the hexagonal structure vs. "x from Rietveld
profile fitting" in LixNi2-xOZ
Fig. 8 The ratio al,/ch multiplied by m v s "x from Rietveld profile fitting".
The struct~ire is cubic for m a l , / c h = l
0 0 .2 0.4 0.6 0.8 1 1.2 x i n Li, Ni,-, 0,
Fig. 9 The long range order parameter from Rietveld profiIe fitting vs. " x
from Rietveld profile fitting". For x<0.62, the data do not indicate long range
order (see text). The solid line is a guide to the eye,
3 ~ 1 n p i c N o X *
I 1 1.032 i1 > 2 0.965 i!)
,. -2 0.889 (1)
4 0.824 (I j
r -7 0.750 (l j
6 0.667 ( 2 )
7 0-608 (3)
e o.sso (I>
3 0.500 (1)
10 0.450 (1)
11 0.400 (I)
12 0.350 (I)
13 0.300 (1)
14 0.250 (1)
15*** 0.200 (I)
a!, (A) 2.8762 (1)
2.8604 (I)
2.8854 (1)
2.8910 (2)
2.8964 (2)
2.9044 (I)
2.9090 (4)
2.9122 (5)
2.9172 (5)
2.9185 (6)
2.9256 (5)
2.9262 (13)
2.9321 (10)
2.9403 (5)
2.94-44 (2)
0 ( z W
0.25S1 (3)
0.2559 (3)
0.2579 (3)
0.2571 (3)
0.2568 (3)
0.2571 (3)
0.2573 (4)
0.2560 (5)
0.2554 (8)
0.2538 (1)
0.2561 (5)
0.2553 (10)'
0.2547 (11)
0.2542 (10)
cubic
(1) QCz) is the z component (parallel to c],) of the oxygen frzctional atomic
csordirrn te.
Lattice Gas lVodel
Tftc sys t tm I,i,Ni2-,02 consists of one FCC frame of oxygen and one FCC
frame uf metal cations. The octahedral interstitial sites of the oxygen lattice
are where the cations can be positioned. At each site, either a Li atom or a Ni
atom can be placed subject to the constraint that we have x Li and (2-x) Ni
a turns per formula unit. A lattice gas model is therefore appropriate for the
treatment of this material. Interactions between metal atoms cause the
ordered arrangement in the ( I l l ) , planes. Fig. 10 shows the structure of the
metal frame, There are twelve nearest neighbors (checkerboard circles) and six
nest nearest neighbors (grey circles) for each site (black circles). All the six
nest nearest neighbors do not share the same ( 1 1 1 ) ~ plane with the site at the
origin. However, six of the nearest neighbors share the same layer with
the atom at the origin and the other six do not.
In the lattice-gas treatment [56J we assume interactions between pairs of
metal cations occupying nearest neighbor (nn) and next nearest neighbor
(nnn) sites and neglect longer ranged interactions. These are denoted JIN*,
J7L"=)71\ ;Lt J I L L for Ni-Ni Ni-Li and Li-Li first neighbor interactions
respectively. Next nearest neighbor interactions use the same notations,
except the subscript 2 replaces the 1. The model Hamiltonian is [561 then
Neares t -Ne ighbo r I n t e r a c t i o n
Figure 10 The cation FCC frame showing the nearest neighbor (checkerboard)
and next-nearest neighbors (grey) to a particular site (black). The first and
second neighbor interaction energies are shown.
I-Icre, xi =I if site i is filled by Li and =O if site i not filled by Li. yj =1 if site i
is filled by Ni and y j =O if site i is filled by Ni. pN and pL are the chemical
pvtcntials for Li and Ni atoms respectively and the sums over the interaction
terms run over all pairs of nearest neighbor atoms (the first term) and all
pairs of next nearest neighbor atoms (the second term).
In our xnodel we assume that all available sites are filled by either Li or Ni
and that there are no cation vacancies. This allows considerable simplification
of our Hamiltonian, since then yi =I - xi. Making this substitution, we obtain
Because we can redefine the zero of cnergy, LVP drop the cunsian! ttlslns h' j tv
N J I N N and T\i]pv" in our subsequent analysis. flrrc, X, is q u a 1 to 1 i t ,I siic is
filled by Li and x; =O if i t is not. Thtreiorc, .rve h;1\.i. t~. ,~nsiortnt~d IIW
Ha~niltonian fro111 that cornmonly used for A-B d loys to t l ~ t u s t ~ l t'or ta l i ticc-
gas models in the trsual fashion 1381.
In order for the (lll)c super-lattice ordering to be stabilized i t tirrns out thlll
one must have J 2 20.5 1 Jz I ; for details the render is refereed to ruicrt l t~~c [-PI].
For this case, a mean-field Brags-Williams treatment of the px-oblcrn is tiscl'tll
which we make below. In the Bragg-Willictlns model, tvc a s s ~ ~ n l t <I
superlattice structure commensurate with the ordered stale. Ass~tining ( I l i ) c
ordering, the Bragg-Williams Hamiltonian (8.4) is obtained by replacing tho
site occupancies xi with s+77/2 and x- q/2 for sites on alternate (11 1 jc mlion
planes. Here x=<xi> is the average lithium concentration in the cation siles,
equal to s in the cornpound Li,Nil-,O and 11 is the order parameter d e l i n d
earlier, in equation (2.1).
Dropping constant terms from (8.2) we have
and we obtain the Bragg-Williams Hamiltonian
H 3 - = p r + 3 ( 2 ~ , + J 2 js' - - j 2 q 2 N 4
irorn iuiz!i i~ rA;c can ~mincdiateiy sce t h t the ~ c a ; > s t neighbor interaction, J7,
tIocls not cct~lyic to the order parameter In mean field theory. Figure 10 shows
this as well; ~i is tr ibut~ng 6 T,i atoms randomly over the 12 nearest neighbors of
n ccntrnl Li atom "custs" GII in energy, the same as if the atoms are in the
orcierecf state, so we do not expect to couple to this ordered state. The order
parxneter 11 is actually four-fold degenerate corresponding to the four cubic -
i t i , 1 1 (1 1 1 1 I ) and ( I i ) with order parameters ql, q2, q 3 and
q4 , rc~peotivtlly.
Does the system select just one q or some linear combination of all four? To
answer this question, we consider the relative entropies for these two
situations. In mean-field theory, entropy is maximized in lattice gas
arrangements with all site occupancies equal to one half, which is not possible
in an ordered state with q>0. When ~ $ 0 , entropy is maximized in ordered
states where the urlzgnitmie of the deviation from half occupancy, is the same
on each site. This certainly true for our ordered state where alternate (111)
planes have average compositions x+ q/l and x-q/2 . In an ordered state
whish is a mixture of the four ( I l l j states, the magnitude or absolute value of
the deviation of the average occupation from 1/2 will vary from site to site.
Hence, the entropy is ~naximized in phases corresponding to one of the four
possible (111) super-lattices and not in a mixture. A detailed proof [56] of this
is given in nypendis one.
The Dragg-Williams free energy is easily shown to be
Expanding in powers of q, w e obtain
Using the Landau theory of phase transitions, we expect that F i v i l l bc
minimized for q =0 whenever
and that F will be minimized for q>0 whenever
This phase transition to q#0 occurs in the Bragg-Williams treatmrnt when
equality holds above, giving
wiicrt: :c, is the critical lithium concentration in LiXNil-,O beyond which
ordtr ing in ( 3 pldnos occurs. The data in Figure 8 shoiv that the ordered
state t l i ~ a p p e ~ m for ~ < ~ , . ; 0 . 6 2 in Li,Ni2_,02 which corresponds to ~ ,=0 .31 in
LiXNil-,O. Subs ti t~rting x,=0.31 into the above equation gives
./, -- = 0.78 klj T (9.6)
Rewriting the Bragg -Williams free energy given by equation (9.1) as
where sn=s+~?/2 and q = x - q f 2 are the site occupancies for sites on alternate
(111)~ ration planes. In equilibrium, the chemical potentials, PA and PB of the
2 sublattices must be equal because they can exchange particles. Therefore
Thus
where Eg is a constant and
These equations are ther! arranged
where
3 d P
E, = - [Eo + - + 3 J 1 (x , + X, ) + 3 J,.\-, ] k,T & 7 (10.6)
and
The equations (10.5-10.7) are easily solved by an iterative method and are used
to find the relation of q versus x and the mean field phase diagram for thc
model.
Figure l l a shows q versus x in LixNi2-x02 for J2/(ksT)=0.78 and for
J2/(k~T)=1.35 (chosen so that the maximum in q agrees with the data)
calculated by minimizing the Bragg-Williams free-energy given by equation
(9.1). This can be directly compared to the data in Figure 9. The agreetnciit is
encouraging and suggests that better solutions to the statistical mechanics,
such as Monte Carlo methods, are in order. Figure l l b shows the mean-field
phase diagram for the model.
Fig. 11 (a) The order parameter versus x calculated by the Bragg-Williams
r~~odr l for the values of jz/ksT shown
(b! The order-disorder phase diagram calculated using the Bragg-
it'illiams 111ocic.l
7 Monte Carlo Results
The Monte Carlo method [50] is nolv coml~~only used in physics. 'The r!;lnlti
Monte Carlo comes from similarity of this method with rodclte ,g~n\cs L h ~ t
use random numbers. The Monte Carlo c net hod does not use thc s~inrc.
method as roulette games to produce the 1O10 required rC-indom nun\bcrs b11t
uses computers to create them in order to save time. The rnnciom nun11wr.s
are then used for simulating configurations of the system.
In thermodynamic fluctuation theory, the distribution function of any
macroscopic variable will be sharply peaked around its average v a l ~ ~ e : c. g. ,
for the energy E itself, the distribution will be peaked a t <E>pr., which is
proportional to N, while the width of the distribution will be proporlioml to
$%. The Ising model is one of the typical samples studied with the Monte
Carlo method. On a scale of energy per spin in Ising model, the width of tho
distribution shrinks to zero as 1/dN for N to infinity. Thus, at any
temperature, a rather narrow region of the configuration space of the SYS~CIII
contributes significantly to the averages; a very small fraction of thc
generated states would actually lie in this important region of configuration
space.
In the Metropolis method, the sample is choscn with a probability
proportional to the Boltzmann factor itself, i.e. , the states arc di5tributcd
according to a Gaussian distribution around the appropriate avoragc vzllue. In
the Ising model example, one starts with some initial spin configurnlion in
Ising r n o d ~ ~ l , Tht.rrnnl fluctuaticjns ;Ire sim~~lateci by repeating the follo~virtg
Gx s t e p aga in and. again:
1) Choose one spin to be considered for flipping (Si to -S,)
2) Calculate the energy change AH relative to that flip.
3) If AI-i.=il, flip the spin, otherwise calculate the transition probability W
dcfincd by W=exp(-~1-I/kp,T) fur that flip
4) Generate a random number y between zero and unity
5) If p<W flip the spin, oiherwise do not flip
6) Calculate the averages as desired with the resulting configuration
In the lattice gas model, each site has two states, x ~ = O (empty) or ~ i = l (filled).
'The Wamiltonian is
If the site is exnpty, xi=.O, and the energy change on filling the site is
If the sitc is filled, xi=l, and the energy change on emptying the site is
The FCC lattice is composed of four simple cubic i a t f i c ~ s anif IS 4 ;\itxns pcr
unit cell Our lattices ranged in size fri711-t a 4sls.f unit ccll to ;t 1 2 x 1 2 ~ 1 2 w i l i l l+
from 256 atoms to 6912 atoms. The boundxy condition in one d i m c n ~ i c ~ n is
where L is the even number of FCC lattice in one dirnertsiar~. I t is impor tmt
to save computer time in this calculation by labelling each neighbor or ncsl
neighbor site by one integer. However, it is much easier to locate a lwighbur
or a next neighbor site by four integers. Figure 12 shows the way to lCtbcl tltc
neighbor and next neighbor sites by four integers.
Figurc 12 The FFC cubic structure is compsed of four simple cubic lattices. Points
on the same simple cubic lattices have the same color.
Anv site can be labelled with the four integers (n, l,, ly, I,), where l<n<4 is
one of the 4 simple cubic lattices in the FCC structure and 1<1,, ly, 1, <L label
the site position on the single cubic lattice. Hence the six next-nearest-
rwighbor sites of the site Cn(O, O,0) are C,(1,0, 3), C,, (-1,0, O), C,(O, I, O), C,(O, -
I. 01, C,,(O, 0, 1) and C,(O, 0, -1). The ~rvelve nearest-neighbors for Cl(0, 0, 0)
are C?(O,O, U), C2(1, OI OIf CzfO, -2,0), Cz(l, -1,0), C3(0,0, O), C3(0, -I,O), C3(0, -1, -
I , 0 , . 0 - 1 0 , 0 0 l C 1 , 0 0 i 0 -1. The twelve nearest
;~t.i@xm for C2iO, 0, 8 ) are Cl(D, 0, d), C1(O, I, O), C1(-I, 0, O), C1(-I, I, O), C3(0, 0,
t)), C$-l,O, O!, C3C-1; 0, -I), C$O, 0, -I), C$(O, 0, O), Q(0, 1,0?, c3(0,0, -11, c4(0, 1, -
Ib. The trrelw nearest neighbors for C$O, 0,O) are C1!0,0, O), Cl(0, I, 01, Cl(0,
We transfer the four nuA ibers into one nun~ ' :~ r for convc~nic.nw in t i t i >
calculation. Any i cai-i be treated as "<l<L undt~r the houndarv c v n i i i t i o ~ ~ .
versus x and T. As order develops, we do not kno~v n-priori ;ilot:g ivhicl~ oi
the 4 equivalent cubic directions (1 1 1 1 1 1 1 1 or 1 1 i) thr order
develops. To calculate the long range order parameter, we calculate the
average site occupancy in iayers nor1 .a1 to (111) and then take the d i f f i ~ r c ~ ~ c ~
between the average occupancies of the ,et of alternate planes. This w e call q,.
We repeat this procedure for planes normal !L- t l x (1 i I ) (1 1 i) and ( 1 I 1)
directions, finding q2, q3 and q: respectivelv. In the ordered state, only one of
the 4 order parameters is significantly different from zero. The over;lll order
parameter, q is then defined to be
to ensure that our computation is sensitive to the devcioprrrent of long range
order in any of the 4 possible directions and to ensure that rj is p s i tivc.
Figure 13 shows 77 verstis M in iixXi2_,U2 calculated for f z =1 .2kr:T with / j =-
O - S ~ B T , for a variety of lattice sizes. At x=I, when the laitice is half filic_:ii by f , i ,
the order parameter reaches a maximum of 0.92, svhirh is zpproxi:nate!l
independent of lattice size. Hoszrever, the order parameter dcpcrrds on the
latiice size near the critical compositiun, s,. The finite sized lattice w e use
~ ~ I c J % ' ' , r c 4 ~ i u a l orcicr Sdaw s,- which decreases as the lattice size increases.
Figtlrc 14 i l ~ i ~ w s versus x for J2 =1.35kr3T calculated b y Monte Carlo (L=12)
and meart iie!d theory for f2=1.35kijT and J2=0.78kBT. We see that mean-field
theory can not f i t the data both at x=l and x,=O.S2 with the same 12.
'The data i n figure 8 suggest that experimentally x,=0.62 and the data in
figure 9 show that the maximum in q at s=I is q,,,,,=0.96. Figure 14 also shows
our attempt io fit the data with the Monte Carlo calculation using f2 =1.35kBT
(11 can be chosen within a wide range without affecting the results
sisnificantly j on the largest (12~12xl2)lattice size considered. The fit is
excelleni in the ordered state which shows that LixNi2-x02 can be well
described by a lattice gas model.
The critical composition, x-, , where order develops, is most determined by
plotting the fluctuations of the order parameter,
versus x . Noise free fluctuation quantities such as <q2> - <q>2 are very
diffirtilt to calculate with conventional Monte Carlo methods, particularly if
successive spin configurations are highly correlated (auto-correlation). In
order to alleviate this problem mentioned in reference [56], we have used
Ferrenberg and Swendsen's Mu1 ti-Hist ogrnm method [51], which allows
optimal use of all the simulation data.
~l~~ yrLltability .-+ .-. -, distribution a: d;emical poteritia! p, PP(x), is generated by
buifding a histsgraln of the number of times the lattice gas configurations
hiwe s in, the range I-; and ~ i + d ~ , where i labels a bin in the histogram and Ax
is the bin size. In this trork ~vc;. used .\.v=-ls 10-5. Fi-om tilt. r . i \ - \vr~i~;ht ing
one can see that the exact probability distribution at ~t i~ctually cor-tl;iins c ~ l l tlxc
necessary information needed to calculate P~,,(ZT) for any other c l ~ c ' n ~ i c ~ ~ l
potential p'. In practice PP(x) only contains information on clistrib~rtions fur p'
near p, because of poor counting stcztistics in the wings of the h is togrm~ i;\r
from However, data quality can be improved considerably by combining
numerous histograms generated at different p's.
In order to obtain an accurate estimate of x, in reference [56], we have
calculated the following three fluctuation quantities
which all have maxima at the phase transition. Figure 1521 shows the
fluctuations in the order parameter q for three different lattice sizes, 1,==4, 8
and 12. Two other lattice sizes, L=6 and 10 (not shown) were also sirnu1atc.d.
The solid lines in figure 1% are calculated using the multiple histugram
method and discrete data points are obtained by standard averaging a t each jr
simulated. For each p, 1000 mcs (Monte Carlo Step) were allowed for the
system to reach equilibrium and a further 10000 rncs for averaging of
thc:rmclt.!yr~a~nic cjuar~tities. I n order to estimate s, in the thermodynamic
limit w e have extrapolated assuming a power law dependence on lattice size
The fits (figure 15b) w a s rather insensitive to the exponent z but the best
results were obtained for x,(m)=0.620(5) and z=1.5.
F i g u r d 6 shows the phase diagram calculated by the Monte Car10 method
with L=12. The phase transition point is taken at the point of maximum
fluctuation of the order parameter. With J1=-0.5k~T, the ordered phase has
the maximum kBT/J2=1.05 at x=I and the minixnuxn at ~ ~ 0 . 4 . Figure 17
shows the order parameters q versus x calculated by the Monte Carlo method
with L-12 and ]1=-0 .5k~T . The order parameter q reaches I at x=l for
]2/k~T>1.6.
Figure 18 shows the effect of J I on the results of q versus x.. With J2=1.35k~T
and L=12, changes in 11, have a negligible effect on q for -0.5kgTcJ1<0.5kBT
and do not effect x, significantly. However, when Jl becomes larger, especially
when Jl>Jz, these changes are significant. At J1=2.0kgTf the ordered phase only
occurs very near x=l.
Fig. 13 The order parameter q from the Monte Carlo calculations for various
lattice sizes and a fixed next-nearest-neighbor interaction constant. The results
are averaged over of 1000 equilibrium ensembles
Fig. 14 Order parameter, q, versus x in LixNi2-x02. The solid points are the
data from figure 9 and tile curves are from the Monte Carlo simulation (L=12)
and the mean field theory
Fig. 15 (a) The fluctuations in the order parameter calculated from Monte
Carlo averages (data points) and the Muiti Histogram lneihod (solid lines) for
3 lattice sizes. Averages and histograms generated f r o 3 10000 Monte C a r b
steps at equilibrium for l2 =135kT and fl=-D.5kT.
(b) Power law (see text) extrapohtion of X, to infinite lattice size giving
~,=0.620(5). 1,'s for finite lattice sizes were calculated from 3 fluctuation
quantities that all have maxima at the phase transiiion.
1-6 1 1 Disorder
0.4 1 A a A Order A A A
Fig. 16 The phase diagram is calculated by Monte Carlo method with L=12
and 11 =-O.S~BT.
x in Li, Ni,, 0,
Fig. 17 The order parameters are calculated by Monte Carlo method for
variety of J2 with L=12 and JI=-O.%BT.
Fig. 18 The order parameters are calculated by Monte Carlo method for
variety of J I 51.i th L=12 and J2=1.35k~T
8 Short-Range Order
Figure 19 shows the (003)H Bragg Peak region 111eas~ired at sevsrd v c d ~ ~ c s 01 :i
in LixNi2-,02 . The data are displayed with different intensity scaXos, btri ihis
does not affect the peak shape. As s decreases below ~4.62, the width of l lw
peak increases rapidly. The broad weak peak indicates the presence o i short-
range order, the precursor to the the long-range order tvhich d c w l o p s hior
X>0.62 as suggested in Figure 9. This short range order rc.presentstlw
tendency of Li atoms to avoid simultaneously filling next-nearest-neiglxbor
sites.
(003) Region 1
1 0 1 5 2 0 2 5 3 0 35 Scattering Angle (Degrees)
Fig. 19 X-ray diffraction shows the short-range order in LiXNi2.,O2 for
0SxS0.62. The Bragg peak in the (003)H region has been multiplied by a scale
factor which increases as x decreases to make the peak visible a t low x
9 Discussion
We have shown that the order-disorder transition in I,i,Ni?-,<) is 1vi.11
modelled by a lattice-gas treatment. Our model reprocfuccs the L I ~ ~ L ~ vcry ~vtllt
using J 2 =l.35kBT for small values of / I (e.g I f l i <0.5kBT ). Thew resrllts
clearly suggest that 12 is larger that J I in this material. This is somc.wh,~t
surprising since the cation-cation bond length corresponding to 12 is 4 2 limes
that corresponding to 11 and usually interaction strengths clccrcnsc: with
distance. However, the interactions 11 and J2 are made up of net Li-Li, Ni-Ni
and Li-Ni interactions as given in equation (8.2), each of these interxtions is
very complicated to estimate.
We have krlow J2=1 .35kBT , and that the X-ray diffraction measurements
reported here have been made at room temperature. It remains to determincj
the value of T (T#Room Temperature). In our opinion, the ilnporlanl
temperature is that at which the Ni atoms can no longer move. For the
lattice gas treatment to be reasonable, both Li and Ni must be mobile and
diffusing freely from site to site. During cooling of our samples the Ni atoms
freeze at some temperature; (probably around 600% since synthesis of I2iNiO2
I 4 is very slow(>40hours) at this temperature); thus our samples ar t s n a p
T i - shots" of the equiiibrium that existed just above that tempcraiure. u5mg
T=6OO OC, we calculate J_7=O.l l et". f reliminary high temperature diffraction
measurements using an ill-sifu furnace show that the ordered phasr? in
C i ~ c gait1 of oiir research ir; to extract the calion-cation interaction energies so
that we ca1-1 jliijdili ihe behavior uf Li/Li,Ni02 batteries with lattice-gas
ini-ideis as h a s been done previously for other intercalation compounds [53,
54j. Cniortunately, v'e hr rive - - shown that the value of the nearest neighbor
irr!eraclirm, ~ t ~ r j s i important for that application, cannot be determined from
the order-disorder i n I.i,Nf~-,U2 since the nearest neighbor interaction is not
sensi!ive to the obser~cd order parameter. Other measurements, such as the
variatiur: n i the Li chemical potential in LixNi2-,O-, with s are needed to
dclerrnim the 1.i-Li contribution to f l .
The T,i transition-metat oxides (Lih402) show a rich variety of structures
based on close packed axygen layers [52]. The Li and 34 atoms form a variety of
ordered arrangi_.ments svhich can probably be explained by suitable choices of
cation-cation interaction energies in a lattice-gas formalism. One of our future
goals is to determine a set of interactions consistent with the observed
structures.
where x is the uniform site occupation. rj is the lattice trcclnr for sitc i, t ~ i ~ i f
-
the qa are the eight ordering wavevectors, ql=(l, 1, I), qz=(l , 1, I), q3=( 1, - - -
I, 11, q4=( I 1 ,I), q5=-qlr q ~ = - q 2 ~ q;.=-q3, q8=-q~. Reality of the s-i inlplics 1 1 1 ~
following constraint
Thus = v X q . and we define = %. + q y q s with similar relationships for flit
other 3 order parameters. Substituting ( A l ) into the Tkndau expnnsirm (9.2)
we obtain
where
which cietcrmiries whether one q, is selected or not. In order to make a
tiic:a~~ingir;i ct~mparison of free energies, the values of the qa must be chosen
so as io satisfy
For sintplicity, we wili test the two extreme cases for which (A6) is satisfied:
Since the coefficient of the fourth order term is positive definite and
jAS)<tA7), the free energy is fower for rase 2 than for case I, i.e. only oize of
the four 1;'s will be selected.
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